A279767 Numbers m such that m and m+2 have the same prime signature.

3, 5, 11, 17, 18, 29, 33, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 107, 137, 141, 143, 149, 159, 179, 183, 185, 191, 197, 201, 203, 213, 215, 217, 219, 227, 235, 239, 242, 247, 248, 265, 269, 281, 299, 301, 303, 306, 311, 319, 321, 327, 339, 340, 347, 348, 391, 393, 411, 413

Offset

1,1

Comments

The sequence contains some terms such that m and m + 2k (k > 1) have the same prime signature. For some terms where m and m + 2k share the same prime signature this means that every alternate element between, and including m and m + 2k have the same prime signature. The first such example is where a(41951) = 402677, a(41953) = 402679, and a(41955) = 402681, share the same prime signature {1, 1}. Also the remaining alternate terms excluding endpoints share the same prime signature. Using the above example, a(41952) = 402678 and a(41954) = 402680 share the prime signature {1,1,3}. - Torlach Rush, Feb 25 2018

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5585 from Michel Marcus)

Index to sequences related to prime signature.

Example

18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.

Mathematica

primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)

Prog

(PARI) isok(n) = vecsort(factor(n)[, 2]) == vecsort(factor(n+2)[, 2]); \\ Michel Marcus, Feb 25 2018

Crossrefs

Cf. A001359, A052213, A052214.

Sequence in context: A323582 A088328 A102643 * A125631 A045408 A092740

Adjacent sequences: A279764 A279765 A279766 * A279768 A279769 A279770

Keyword

nonn,easy

Author

Altug Alkan, Dec 18 2016

Status

approved